5 Week Modular Course in Statistics & Probability Strand 1. Module 2

Transcription

1 5 Week Modular Course in Statistics & Probability Strand 1 Module 2

2 Analysing Data Numerically Measures of Central Tendency Mean Median Mode Measures of Spread Range Standard Deviation Inter-Quartile Range Module 2.1 Mean & Standard Deviation using a Calculator Calculate the mean and standard deviation of the following 10 students heights by: (i) using the data unsorted (ii) creating a frequency table Unsorted Data Frequency Table Gender Height/cm 1. Boy Girl Boy Boy Girl Boy Girl Girl Boy Boy 179 Height/cm Frequency Mean = cm S.D. = 9.32cm Module 2.2

3 Central Tendency: The Mean Advantages: Mathematical centre of a distribution Does not ignore any information Disadvantages: Influenced by extreme scores and skewed distributions May not exist in the data Module 2.3 Central Tendency: The Mode Advantages: Good with nominal data Easy to work out and understand The score exists in the data set Disadvantages: Small samples may not have a mode More than one mode might exist Module 2.4

4 Central Tendency: The Median Advantages: Not influenced by extreme scores or skewed distribution Good with ordinal data Easier to calculate than the mean Considered as the typical observation Disadvantages: May not exist in the data Does not take actual data into account only its (ordered) position Difficult to handle theoretically Module 2.5 Summary: Relationship between the 3 M s Characteristics Mean Median Mode Consider all the data in calculation Yes No No Easily affected by extreme data Yes No No Can be obtained from a graph No Yes Yes Should be one of the data No No Yes Need to arrange the data in ascending order No Yes No Module 2.6

5 Using Appropriate Averages Example There are 10 house in Pennylane Close. On Monday, the numbers of letters delivered to the houses are: Calculate the mean, mode and the median of the number of letters. Comment on your results. Solution Mean = 10 = 5.1 Mode = 0 Median = 2 In this case the mean (5.1) has been distorted by the large number of letters delivered to one of the houses. It is, therefore, not a good measure of a 'typical' number of letters delivered to any house in Pennylane Close. The mode (0) is also not a good measure of a 'typical' number of letters delivered to a house, since 7 out of the 10 houses do acutally receive some letters. The median (2) is perhaps the best measure of the 'typical' number of letters delivered to each house, since half of the houses received 2 or more letters and the other half received 2 or fewer letters. Module 2.7 Quartiles Second 25% of Data Third 25% of Data First 25% of Data Last 25% of Data Q1 Q2 Q3 Quartiles When we arrange the data is ascending order of magnitude and divide them into four equal parts, the values which divide the data into four equal parts are called quartiles. TheyareusuallydenotedbythesymbolsQ 1 isthelowestquartile(orfirstquartile)where 25% of the data lie below it; Q 2 is the middle quartile ( or second quartile or median) where50%ofthedataliebelowit; Q 3 istheupperquartile(orthirdquartile)where75% ofthedataliebelowit. Module 2.8

6 Interquartile Range & Range Data in ascending order of magnitude First 25% of data Second25%of data Third25%of data Last 25% of data Q1 Lower Quartile Interquartile Range Q2 Median Q3 Upper Quartile Minimum Maximum Range Module 2.9 Standard Deviation Example Two machines A and B are used to measure the diameter of a washer. 50 measurements of a washer are taken by each machine. If the standard deviations of measurements taken by machine A and B are 0.4mm and 0.15mm respectively, which instrument gives more consistent measurements? Solution Standard deviation of A = 0.4 mm Standard deviation of B = 0.15 mm The smaller the standard deviation, the less widely dispersed the data is. This means that more measurements are closer to the mean. Therefore, the measurements taken by instrument B are more consistent. Module 2.10

7 Guide to Distributions Example 1 Seven teenagers at a youth club were asked their age. They gave the following ages: 16, 14, 19, 16, 13, 18, 16 The mean, mode and the median of their ages are as follows: Mean = 16 Mode = 16 Median = 16 If a line plot of their ages is drawn we get the following Mean, Mode and Median When the Meanand Medianare the same value the plot is symmetrical (Bell Shaped). The Modeaffects the height of the of the Bell Shaped curve. Module 2.11 Example 2: Seven people were asked how many text messages they send (on average) everyweek. The results were as follows: 3, 25, 30, 30, 30, 33, 40 The mean mode and the median of text messages sent are as follows: Mean = Mode = 30 Median = 30 If a line plot of the number of texts sent is drawn we get the following: Mean Mode Median When the Meanis to the left of the Medianthe data is said to be skewed to the leftor negatively skewed. The Modeaffects the height of the curve. Module 2.12

8 Example 3: Eight factory workers were asked to give their annual salary. The results are as follows: (figures in thousands of Euro) 20, 22, 25, 26, 27, 27, 70 The mean mode and the median of their annual salaries are as follows: Mean = 31 Mode = 27 Median = 26 If a line plot of their salaries is drawn we get the following Median Mode Mean When the Meanis to the rightof the Medianthe data is said to be skewed to the rightor positively skewed. The Modeaffects the height of the curve. Module Involves 2 variables Bivariate Data 2. Deals with causes or relationships 3. The major purpose of bivariateanalysis is to determine whether relationships exist We will look at the following: Scatter plots Correlation Correlation coefficient Correlation & causality Line of Best Fit Correlation coefficient not equal to slope Sample question: Is there a relationship between the scores of students who study Physics and their scores in Mathematics? Module 2.14

9 Univariate Data versus Bivariate Data Univariate data: Only one item of data is collected e.g. height Bivariate Data: Data collected in pairs to see if there is a relationshipbetween the variables e.g. height and arm span, mobile phone bill and age etc. Examples: Categorical paired data: Discrete paired: Continuous paired: Colour of eyes and gender Number of bars eaten per week and number of tooth fillings Height and weight Category and discrete paired: Type of dwelling and number of occupants etc. [Look at C@S questionnaire] Module 2.15 Correlation Correlation:is about assessing the strength of the relationship between pairs of data. The first step in determining the relationship between 2 variables is to draw a Scatter Plot. After establishing if a Linear Relationship (Line of Best Fit) exists between 2 variables X and Y, the strength of the relationship can be measured and is known as the correlation coefficient (r). Correlation is a precise term describing the strength and direction of the linear relationship between quantitative variables. Module 2.16

10 Correlation Coefficient (r) 1 r 1 r=+1 Corresponds to a perfect positive linear correlation where the points lie exactly on a straight line [The line will have positive slope] r=0: Correponds to little or no correlation i.e. as x increases there is no definite tendency for the values of y to increase or decrease in a straight line r= 1: Correponds to a perfect negative linear correlation where the points lie exactly on a straight line [The line will have negative slope] r close to +1: Indicates a strong positive linear correlation, i.e. y tends to increase as x increases r close to 1: Indicates a strong negative linear correlation, i.e. y tends to decrease as x increases The correlation coefficient (r) is a numerical measure of the direction and strength of a linear association. Module 2.17 Scatter Plots Can show the relationship between 2 variables using ordered pairs plotted on a coordinate plane The data points are not joined The resulting pattern shows the type and strength of the relationship between the two variables Where a relationship exists, a line of best fit can be drawn (by eye) between the points Scatter plots can show positive or negative correlation, weak or strong correlation, outliers and spread of data An outlier is a data point that does not fit the pattern of the rest of the data. There can be several reasons for an outlier including mistakes made in the data entry or simply an unusual value. Module 2.18

11 Describing Correlation Form: Straight, Curved, No pattern Direction: Positive, Negative, Neither Strength: Weak, Moderate, Strong Unusual Features: Outliers, Subgroups Exam Results Golf Scores High I.Q. Study time Practice time Height Positive correlation As one quantity increases so does the other Negative correlation As one quantity increases the other decreases. No correlation Both quantities vary with no clear relationship Module 2.19 Line of Best Fit Roughly goesthrough the middle of the scatter of the points To describe it generally: it has aboutas many points on one side of the line as the other, and it doesn t have to go through any of the points It can go through some, all or none of the points Strong correlation is when the scatter points lie very close to the line It also depends on the size of the sample from which the data was chosen Strong positive correlation Moderate positive correlation No correlation no linear relationship Moderate negative correlation Strong negative correlation Module 2.20

12 Example A set of students sat their mock exam in English, they sat their final exam in English at a later date. The marks obtained by the students in both examinations were as follows: Students A B C D E F G H Mock Results Final Results (a) (b) Draw a Scatter Plot for this data and draw a line of best fit. Is there a correlation between mock results and final results? Solution (a) (b) There is a positive correlation between mock results and final results. Module 2.21 Correlation Coefficient by Calculator r = a = b = y = x Before doing this on the calculator, the class should do a scatter plot using the data in the table. Discuss the relationship between the data (i.e. grams of fat v calories). Module 2.22

13 Correlation versus Causation Correlation is a mathematical relationship between 2 variables which are measured A Correlation of 0, means that there is no linear relationship between the 2 variables and knowing one does not allow prediction of the other Strong Correlation may be no more than a statistical association and does not imply causality Just because there is a strong correlation between 2 things does not mean that one causes the other. A consistently strong correlation may suggest causation but does not prove it. Look at these examples which show a strong correlation but do not prove causality: E.g. 1 With the decrease in the number of pirates we have seen an increase in global warming over the same time period. Does this mean global warming is caused by the decrease in pirates? E.g.2With the increase in the number of television sets sold an electrical shop has seen an increase in the number of calculators sold over the same time period. Does this mean that buying a television causes you to buy a calculator? Module 2.23 Criteria for Establishing Causation There has to be a strong consistent association found in repeated studies The cause has to be plausible and precede the effect in time Higher doses will result in stronger responses Module 2.24

14 Correlation & Slope undefined Several sets of (x,y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the spread and direction of a linear relationship but not the gradient (slope) of that relationship, N.B.: the figure in the centre of the second linehasaslopeof0but inthatcasethecorrelationcoefficientisundefined becausethe varianceofyiszero. The gradient (slope) of the line of best fit is not important when dealing with correlation, except that a vertical or horizontal line of best fit means that the variables arenotconnected.[thesignoftheslopeofthelineofbestfitwillbethesameasthatof the correlation coefficient because both will be in the same direction.] Module 2.25 Notes Module 2.26